Problem: Determine how many solutions exist for the system of equations. ${5x+y = 1}$ ${y = 1-5x}$
Solution: Convert both equations to slope-intercept form: ${5x+y = 1}$ $5x{-5x} + y = 1{-5x}$ $y = 1-5x$ ${y = -5x+1}$ ${y = 1-5x}$ ${y = -5x+1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x+1}$ ${y = -5x+1}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${5x+y = 1}$ is also a solution of ${y = 1-5x}$, there are infinitely many solutions.